The fact that electrons have possessed quantized angular momenta has been known for many years. Classical experiments, such as the Stern-Gerlach experiment, have provided experimental observations of this property. In this experiment, silver atoms were ejected from a furnace and onto a detector, passing through a magnetic field perpendicular to the direction of atom movement along the way. The d9-silver atoms possess one unpaired electron, allowing for ease in evaluating of the magnetic properties of the paramagnetic atom. The results of the experiment showed that the silver atom beam separated into two distinct beams, equally and oppositely affected by the magnetic field, with no atoms on the detector between the beams. This observation corroborates experimental results from other phenomena (ex. doublet spectral lines in hydrogen atom from spin-orbit interactions) and confirms the presence of a quantization of
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angular momentum that could only be described by the unpaired electron on the silver atom. Ultimately, these showed that electrons possessed two angular momenta, or spins, of equal and opposite magnitude.1
Most uses of electrical currents ignore the effects of electron spin in their execution. The effects of spin can be observed in currents, such as the spin Hall effect leading to accumulation of spin on the lateral boundaries of a current-carrying sample,2 but practically these effects bear little consequence on moving current (the current flowing through an electrical outlet is
effectively spin randomized), it only matters whether or not electrons of any spin are transported. New avenues are opened when electron spin is incorporated, and even the simple picture of current on versus current off can be manipulated in spintronic settings.
The basis for almost all, if not all, the magnetic effects in spintronics comes from energy splitting of angular momenta in a magnetic field:
𝛥𝐸 = −𝜇 ∙ 𝐵 (2-1)
where 𝜇 is the magnetic moment and B is the applied magnetic field. Since electron spin will inherently contribute a magnetic moment, any application of a field will create an energy split where one spin state will become more energetically favorable than another. To give a more numerical description to this splitting, we define 𝜇 by visualizing an electron as a charge moving circularly around a nucleus as predicted in the Bohr atomic picture. The basic definition for magnetic moment defines it as:
𝜇 = 𝐼𝐴 (2-2)
where I is current and A is area of the current. In the Bohr atomic picture, A is a circle defined by A = πr2. The circulating current, I, is given by:
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𝐼 = −𝑒 𝑇 =
𝑒𝑣
2𝜋𝑟 (2-3)
where the period, T, is simply expanded to represent the electron moving around the nucleus. If we remember that the electron and its energy (and therefore momentum) are quantized as per basic quantum mechanics principles, we can apply the definition:
𝐿 = 𝑚𝑣𝑟 =𝑛ℎ2𝜋 (2-4)
where L is angular momentum, h is Planck constant, and n represents an integer quantization of that energy. Using the equations above, we can combine them to obtain the magnetic moment for a n = 1 system:
𝜇 = 𝑒ℎ
4𝜋𝑚= 𝜇𝐵 (2-5)
We plug in known values for the charge and mass of an electron and we obtain the quantity known as the Bohn magneton (𝜇B = 9.27 × 10-24 Am2). This number is interesting, coming from a now known to be erroneous picture of the atom, yet still representing a quantized magnetic moment that small scale magnetic systems are scaled to.3 It also gives credence to the idea that consequential magnetic effects, especially those at the single molecule level, can come simply by considering the spin on an electron.
This single electron picture, while useful, quickly becomes complicated when we consider electron-electron interactions. These exchange interactions differ between systems, enabling spin alignment or spin anti-alignment under different circumstances. A basic equation to express this interaction is given by the Heisenberg-Dirac-Van Vleck (HDVV) Hamiltonian:
𝐻̂𝑖𝑗 = −2𝐽𝑖𝑗𝑆̂𝑖𝑆̂𝑗 (2-6)
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In this basic relationship, the magnitudes of the spins are proportional to the magnetic effect, and the exchange parameter term accounts for all interactions to determine ground state spin
preference. Using the exchange parameter in this manner can make it difficult to fully dissect, but it still allows us to comment on two general but widespread classes of interaction –
ferromagnetic coupling, where Jij > 0 and it is energetically preferable for the spins to align; and antiferromagnetic coupling, where Jij < 0 and it is energetically preferable for the spins to be anti-aligned. Again, more complex interactions, such as superexchange and double exchange can very easily make this picture complicated.4 However, what this model expresses is that different spin interactions exist, some where spins will align with each other, others where spins will anti- align.
In metals, both types of coupling are well represented. Ferromagnetic coupling leads to a few classes of magnetic materials, paramagnets and ferromagnets being the most common of them. Both have the same property where spins will align in a magnetic field. The difference is that the spin state becomes randomized in paramagnetic species when the magnetic field is removed, whereas ferromagnets will retain a remnant magnetization (Figure 2.1). The remnant magnetization, alongside hysteresis of spin flipping, allows ferromagnets to be useful for magnetic memory. Over time, thermal fluctuations will cause spin randomization in
ferromagnets, with each ferromagnet possessing a characteristic Curie temperature (TC) above which they lose their remnant magnetization, instead becoming what are known as
superparamagnets. Superparamagnets can become ferromagnetic when T > TC, however any previous spin state from a previous ferromagnetic state is still lost.
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Figure 2.1 – Magnetism in Manganese-Doped Carbon Nanotubes (CNTs)
Ferromagnetism expressed in manganese doped CNTs (“%” refer to percent Mn doping), measured at low temperature (10 K), as shown through hysteresis of magnetic signal and remnant magnetization at 0 Oe fields. Reprinted with permission from Ref. 5. Copyright 2012, AIP Publishing LLC.
Antiferromagnetic coupling can lead to antiferromagnets, where spins are spin anti- aligned with each other throughout a sample. Antiferromagnetism can be common in some transition metals, specifically oxides of those metals. As with ferromagnets, antiferromagnets can lose their magnetic properties due to thermal fluctuation above the Néel temperature (TN). One of the interesting uses of antiferromagnets is that they can pin the magnetism at a ferromagnetic interface, creating larger coercive fields for spin flipping. Apart from this, they will ultimately resist the effects of magnetic field, acting essentially as strong diamagnets and can be repelled from a magnetic field.
These basic magnetic effects have been widely utilized in a variety of applications. Magnetic memory builds off of spin valve configurations where two ferromagnets, separated by an insulating tunnel barrier, control the resistance of a device based on their spin alignment or anti-alignment,6 seeing sometimes exceptionally large MR effects in specialized systems.7 Diamagnetism and its repulsion to magnetic field can lead to levitation of any number of
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materials.8 Sensors, magnetic resonance imaging (MRI), and some revered characterization techniques in chemistry such as nuclear magnetic resonance (NMR) are among the many other uses of the principles discussed above.
Figure 2.2 – Levitating Frog in a 20 Tesla Magnet
Levitation of a diamagnetic frog in the bore of a strong magnet, exhibiting the general ability to levitate diamagnetic materials due to their repulsion to a magnetic field.8 © European Physical Society. Reproduced by permission of
IOP Publishing. All rights reserved.